scholarly journals Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1803
Author(s):  
Dmitry Efrosinin ◽  
Natalia Stepanova ◽  
Janos Sztrik ◽  
Andreas Plank
Keyword(s):  
Optimal Policy ◽  
Optimal Allocation ◽  
Queueing System ◽  
Control Policy ◽  
Iteration Algorithm ◽  
Heuristic Solution ◽  
Allocation Policy ◽  
Markov Decision Model ◽  
The Mean ◽  
Number Of Customers

The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions.

scholarly journals Analysis of the Optimal Resource Allocation for a Tandem Queueing System

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Zaiming Liu ◽  
Wei Deng ◽  
Gang Chen
Keyword(s):  
Optimal Policy ◽  
Optimal Allocation ◽  
Queueing System ◽  
Service Rate ◽  
Allocation Policy ◽  
Long Run ◽  
Markov Decision ◽  
Upstream Station ◽  
Optimal Resource ◽  
Resource Cost

We study a controllable two-station tandem queueing system, where customers (jobs) must first be processed at upstream station and then the downstream station. A manager dynamically allocates the service resource to each station to adjust the service rate, leading to a tradeoff between the holding cost and resource cost. The goal of the manager is to find the optimal policy to minimize the long-run average costs. The problem is constructed as a Markov decision process (MDP). In this paper, we consider the model in which the resource cost and service rate functions are more general than linear. We derive the monotonicity of the optimal allocation policy by the quasiconvexity properties of the value function. Furthermore, we obtain the relationship between the two stations’ optimal policy and conditions under which the optimal policy is unique and has the bang-bang control property. Finally, we provide some numerical experiments to illustrate these results.

On a property of the variance of the waiting time of a queue

1968 ◽  
Vol 5 (3) ◽  
pp. 702-703 ◽  
Author(s):  
D. G. Tambouratzis
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
The Mean ◽  
Number Of Customers

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.

scholarly journals Estimation of the Optimal Threshold Policy in a Queue with Heterogeneous Servers Using a Heuristic Solution and Artificial Neural Networks

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1267
Author(s):  
Dmitry Efrosinin ◽  
Natalia Stepanova
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Optimal Allocation ◽  
Optimal Threshold ◽  
Iteration Algorithm ◽  
Heterogeneous Servers ◽  
Heuristic Solution ◽  
Threshold Policy ◽  
Optimal Thresholds ◽  
Artificial Neural

This paper deals with heterogeneous queues where servers differ not only in service rates but also in operating costs. The classical optimisation problem in queueing systems with heterogeneous servers consists in the optimal allocation of customers between the servers with the aim to minimise the long-run average costs of the system per unit of time. As it is known, under some assumptions the optimal allocation policy for this system is of threshold type, i.e., the policy depends on the queue length and the state of faster servers. The optimal thresholds can be calculated using a Markov decision process by implementing the policy-iteration algorithm. This algorithm may have certain limitations on obtaining a result for the entire range of system parameter values. However, the available data sets for evaluated optimal threshold levels and values of system parameters can be used to provide estimations for optimal thresholds through artificial neural networks. The obtained results are accompanied by a simple heuristic solution. Numerical examples illustrate the quality of estimations.

On a property of the variance of the waiting time of a queue

1968 ◽  
Vol 5 (03) ◽  
pp. 702-703 ◽  
Author(s):  
D. G. Tambouratzis
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
The Mean ◽  
Number Of Customers

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.

Analysis of a non-preemptive priority multiserver queue

1988 ◽  
Vol 20 (04) ◽  
pp. 852-879 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Probability Distribution ◽  
Queueing System ◽  
Waiting Times ◽  
Complex Variables ◽  
Arrival Process ◽  
Preemptive Priority ◽  
Multiple Servers ◽  
The Mean ◽  
Equilibrium Probability ◽  
Number Of Customers

We consider a non-preemptive priority head of the line queueing system with multiple servers and two classes of customers. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. A Markovian state description consists of the number of customers of each class in service and in the queue. We solve a matrix equation to obtain the generating function of the equilibrium probability distribution by analyzing singularities of the equation coefficients, which are meromorphic matrices of two complex variables. We then obtain the mean waiting times for each class.

scholarly journals Analysis of an M|G|1|R queue with batch arrivals and two hysteretic overload control policies

2014 ◽  
Vol 24 (3) ◽  
pp. 519-534 ◽  
Author(s):  
Yuliya Gaidamaka ◽  
Alexander Pechinkin ◽  
Rostislav Razumchik ◽  
Konstantin Samouylov ◽  
Eduard Sopin
Keyword(s):  
Queueing System ◽  
Overload Control ◽  
Control Policies ◽  
Batch Arrivals ◽  
Hysteretic Control ◽  
The Mean ◽  
Sip Server ◽  
System Mode ◽  
Number Of Customers ◽  
Mean Time

Abstract Hysteretic control of arrivals is one of the most easy-to-implement and effective solutions of overload problems occurring in SIP-servers. A mathematical model of an SIP server based on the queueing system M[X]|G|1(L,H)|(H,R) with batch arrivals and two hysteretic loops is being analyzed. This paper proposes two analytical methods for studying performance characteristics related to the number of customers in the system. Two control policies defined by instants when it is decided to change the system’s mode are considered. The expression for an important performance characteristic of each policy (the mean time between changes in the system mode) is presented. Numerical examples that allow comparison of the efficiency of both policies are given

Analysis of a non-preemptive priority multiserver queue

1988 ◽  
Vol 20 (4) ◽  
pp. 852-879 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Probability Distribution ◽  
Queueing System ◽  
Waiting Times ◽  
Complex Variables ◽  
Arrival Process ◽  
Preemptive Priority ◽  
Multiple Servers ◽  
The Mean ◽  
Equilibrium Probability ◽  
Number Of Customers

We consider a non-preemptive priority head of the line queueing system with multiple servers and two classes of customers. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. A Markovian state description consists of the number of customers of each class in service and in the queue. We solve a matrix equation to obtain the generating function of the equilibrium probability distribution by analyzing singularities of the equation coefficients, which are meromorphic matrices of two complex variables. We then obtain the mean waiting times for each class.

scholarly journals Analysis of MAP/PH(1), PH(2)/2 Queue with Bernoulli Vacations

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
B. Krishna Kumar ◽  
R. Rukmani ◽  
V. Thangaraj
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing System ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Steady State Probability ◽  
Matrix Geometric Method ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers

We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.

On a Markovian queue with weakly correlated interarrival times

1981 ◽  
Vol 18 (01) ◽  
pp. 190-203 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Interarrival Time ◽  
Probability Vector ◽  
Common Value ◽  
Markovian Queue ◽  
The Stability ◽  
Number Of Customers ◽  
Stationary Probabilities ◽  
Stationary Probability Vector

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated. The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

OPTIMAL CONTROL OF FLEXIBLE SERVERS IN TWO TANDEM QUEUES WITH OPERATING COSTS

2007 ◽  
Vol 22 (1) ◽  
pp. 107-131 ◽  
Author(s):  
Dimitrios G. Pandelis
Keyword(s):  
Optimal Control ◽  
Optimal Policy ◽  
Optimal Allocation ◽  
Operating Costs ◽  
Queuing Systems ◽  
Tandem Queues ◽  
Holding Costs ◽  
Flexible Servers ◽  
Tandem Queuing ◽  
Dedicated Servers

We consider two-stage tandem queuing systems with dedicated servers in each station and flexible servers that can serve in both stations. We assume exponential service times, linear holding costs, and operating costs incurred by the servers at rates proportional to their speeds. Under conditions that ensure the optimality of nonidling policies, we show that the optimal allocation of flexible servers is determined by a transition-monotone policy. Moreover, we present conditions under which the optimal policy can be explicitly determined.

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